R-layer Mode Theory (RLMT) LIII: R-layer Holography and the Quantum Information Structure of Cosmological Spacetime

This volume develops the holographic and quantum‑informational sector of the R-layer Mode Theory (RLMT). We construct a cosmological perturbation theory for the R-layer field, embedding modular flow, entanglement entropy, and coarse‑graining layers into a perturbed FLRW universe. The R-layer acts as an effective quantum‑informational degree of freedom whose dynamics modifies scalar, vector, and tensor perturbations, leading to distinctive signatures in the CMB, large‑scale structure, and gravitational waves. We formulate the R-layer action, derive its stress tensor, obtain gauge‑invariant perturbation variables, and present the Mukhanov–Sasaki equation for the R-layer. We then connect these results to holographic principles, modular Hamiltonians, entanglement wedges, and double‑holographic constructions. This volume establishes the quantum‑informational foundation of RLMT and provides a testable framework for probing the quantum structure of spacetime through cosmological observables.